(x-2).jpeg "(x-3)(x-2)")
Understanding (x-3)(x-2)
The expression (x-3)(x-2) is a product of two binomials. Understanding this expression involves several key aspects:
Expanding the Expression
To understand what the expression represents, we can expand it using the distributive property (also known as FOIL method):
- First: x * x = x²
- Outer: x * -2 = -2x
- Inner: -3 * x = -3x
- Last: -3 * -2 = 6
Adding all these terms together, we get: x² - 2x - 3x + 6
Simplifying this, we have the expanded form: x² - 5x + 6
Factoring the Expression
The expanded form, x² - 5x + 6, is a quadratic expression. We can factor it back into its original binomial form:
- Find two numbers that add up to -5 (the coefficient of the x term) and multiply to 6 (the constant term).
- These numbers are -3 and -2.
- Therefore, we can rewrite the expression as: (x - 3)(x - 2)
Solving for x
The expression (x-3)(x-2) represents a quadratic equation. To find the values of x that satisfy the equation, we set it equal to zero:
(x-3)(x-2) = 0
This equation is true when either (x-3) = 0 or (x-2) = 0. Solving for x in each case, we get:
- x = 3
- x = 2
These are the roots of the equation, meaning they are the values of x where the expression equals zero.
Graphing the Expression
The expression (x-3)(x-2) represents a parabola when graphed. The roots of the equation (x = 3 and x = 2) correspond to the x-intercepts of the parabola. The parabola opens upwards because the coefficient of the x² term is positive.
Applications
The expression (x-3)(x-2) can be used in various applications, including:
- Solving quadratic equations
- Finding the roots of a polynomial function
- Modeling real-world scenarios involving quadratic relationships
By understanding how to expand, factor, solve, and graph this expression, we gain valuable insights into the world of quadratic equations and their applications.